A152668 Number of runs of even entries in all permutations of {1,2,...,n} (the permutation 274831659 has 3 runs of even entries: 2, 48 and 6).

2, 6, 36, 192, 1440, 10800, 100800, 967680, 10886400, 127008000, 1676505600, 22992076800, 348713164800, 5492232345600, 94152554496000, 1673823191040000, 32011868528640000, 633834996867072000, 13380961044971520000

Offset

2,1

Comments

a(n) = Sum(k*A152667(n,k), k=1..floor(n/2)).

Links

Table of n, a(n) for n=2..20.

Formula

a(2n) = (n+1)(2n)!/2;

a(2n+1) = n(n+2)(2n)!.

D-finite with recurrence a(n) -2*a(n-1) -n*(n-1)*a(n-2) +2*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 24 2022

Example

a(3) = 6 because each of the permutations 123, 132, 213, 231, 312, 321 has exactly 1 run of even entries.

Maple

ae := proc (n) options operator, arrow: (1/2)*factorial(2*n)*(n+1) end proc: ao := proc (n) options operator, arrow: n*(n+2)*factorial(2*n) end proc: a := proc (n) if `mod`(n, 2) = 0 then ae((1/2)*n) else ao((1/2)*n-1/2) end if end proc; seq(a(n), n = 2 .. 20);

Mathematica

a[n_] := If[EvenQ[n], (n/2+1)n!/2, ((n-1)/2)((n-1)/2+2)(n-1)!];
Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Apr 09 2024 *)

Crossrefs

Cf. A152667, A052618.

Sequence in context: A369080 A101609 A281024 * A213918 A239889 A086325

Adjacent sequences: A152665 A152666 A152667 * A152669 A152670 A152671

Keyword

nonn,changed

Author

Emeric Deutsch, Dec 14 2008

Status

approved